A304452 - OEIS

%I #4 May 16 2018 20:05:03

%S 1,3,5,6,7,10,11,13,14,16,18,20,22,24,25,26,27,29,31,32,34,36,37,40,

%T 42,44,45,46,49,50,51,53,55,56,58,59,61,62,63,65,67,68,71,72,73,76,78,

%U 80,81,84,85,86,89,90,91,94,96,98,99,100,103,104,105,106

%N Solution (b(n)) of the complementary equation in Comments.

%C Define sequences a(n) and b(n) recursively, starting with b(0) = 1:

%C b(n) = least new;

%C a(n) = b(n) + b(floor(n/2)),

%C where "least new k" means the least positive integer not yet placed.

%C ***

%C Conjectures: a(n)/n -> 5/2 and -1 <= 5/2 - a(n)/n <= 2 for n >= 1;

%C b(n)/n -> 5/3 and -1 <= 5/3 - b(n)/n <= 2 for n >= 1.

%e a(0) = b(0) + b(0) = 2;

%e a(1) = b(1) + b(2) >= 3 + 4, so that b(2) = 3, b(2) = 4, b(3) = 5, b(4) = 6, and a(1) = 7.

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t z = 1000; a = {}; b = {1};

%t Do[AppendTo[a, Last[b] + b[[Floor[(1 + Length[b])/2]]]];

%t   AppendTo[b, mex[Flatten[{a, b}], 1]], {z}];

%t Take[a, 100]  (* A304451 *)

%t Take[b, 100]  (* A304452 *)

%t (* _Peter J. C. Moses_, May 10 2018 *)

%Y Cf. A304451.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, May 16 2018